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https://journals.aps.org/prd/abstract/10.1103/PhysRevD.99.123505
DOI: 10.1103/PhysRevD.99.123505
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by American Physical Society. All rights reserved.
DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
Dark Energy Survey year 1 results: Constraints on extended cosmological
models from galaxy clustering and weak lensing
T. M. C. Abbott,1 F. B. Abdalla,2,3 S. Avila,4 M. Banerji,5,6 E. Baxter,7K. Bechtol,8M. R. Becker,9 E. Bertin,10,11 J. Blazek,12,13S. L. Bridle,14D. Brooks,2 D. Brout,7D. L. Burke,15,16A. Campos,17,18 A. Carnero Rosell,19,20 M. Carrasco Kind,21,22J. Carretero,23F. J. Castander,24,25 R. Cawthon,26C. Chang,26A. Chen,27M. Crocce,24,25 C. E. Cunha,15L. N. da Costa,20,19C. Davis,15J. De Vicente,28J. DeRose,29,15S. Desai,30E. Di Valentino,14H. T. Diehl,31
J. P. Dietrich,32,33S. Dodelson,18P. Doel,2 A. Drlica-Wagner,31T. F. Eifler,34,35J. Elvin-Poole,14,12 A. E. Evrard,36,27 E. Fernandez,23A. Fert´e,37,38,39B. Flaugher,31P. Fosalba,25,24J. Frieman,31,26J. García-Bellido,40E. Gaztanaga,25,24
D. W. Gerdes,27,36T. Giannantonio,6,5,41D. Gruen,15,16R. A. Gruendl,21,22J. Gschwend,20,19G. Gutierrez,31 W. G. Hartley,2,42D. L. Hollowood,43K. Honscheid,12,44B. Hoyle,45,41D. Huterer,27B. Jain,7 T. Jeltema,43 M. W. G. Johnson,22M. D. Johnson,22A. G. Kim,46E. Krause,35K. Kuehn,47N. Kuropatkin,31O. Lahav,2 S. Lee,44,12
P. Lemos,5,6 C. D. Leonard,18T. S. Li,26,31A. R. Liddle,37M. Lima,19,48H. Lin,31M. A. G. Maia,19,20J. L. Marshall,49 P. Martini,50,12F. Menanteau,22,21C. J. Miller,27,36R. Miquel,23,51V. Miranda,35J. J. Mohr,45,52,53J. Muir,15R. C. Nichol,4
B. Nord,31R. L. C. Ogando,20,19 A. A. Plazas,34 M. Raveri,26R. P. Rollins,14A. K. Romer,54A. Roodman,15,55 R. Rosenfeld,19,56S. Samuroff,18E. Sanchez,28V. Scarpine,31R. Schindler,55M. Schubnell,27D. Scolnic,26L. F. Secco,7 S. Serrano,25,24I. Sevilla-Noarbe,28M. Smith,57M. Soares-Santos,58F. Sobreira,19,59E. Suchyta,60M. E. C. Swanson,22
G. Tarle,27D. Thomas,4 M. A. Troxel,12,44 V. Vikram,9 A. R. Walker,1 N. Weaverdyck,27R. H. Wechsler,55,29,15 J. Weller,53,45,41 B. Yanny,31Y. Zhang,31and J. Zuntz37
(DES Collaboration)
1
Cerro Tololo Inter-American Observatory,
National Optical Astronomy Observatory, Casilla 603, La Serena, Chile
2
Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, United Kingdom
3
Department of Physics and Electronics, Rhodes University, P.O. Box 94, Grahamstown, 6140, South Africa
4
Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom
5Kavli Institute for Cosmology, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, United Kingdom
6Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, United Kingdom
7Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA
8LSST, 933 North Cherry Avenue, Tucson, Arizona 85721, USA 9
Argonne National Laboratory, 9700 South Cass Avenue, Lemont, Illinois 60439, USA
10CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France 11
Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
12
Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, Ohio 43210, USA
13
Institute of Physics, Laboratory of Astrophysics, École Polytechnique F´ed´erale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
14
Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
15
Kavli Institute for Particle Astrophysics & Cosmology, P.O. Box 2450, Stanford University, Stanford, California 94305, USA
16
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
17Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil 18
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
19Laboratório Interinstitucional de e-Astronomia—LIneA, Rua Gal. Jos´e Cristino 77, Rio de Janeiro,
RJ—20921-400, Brazil
20Observatório Nacional, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ—20921-400, Brazil 21
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, Illinois 61801, USA
22National Center for Supercomputing Applications, 1205 West Clark Street, Urbana, Illinois 61801, USA 23
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
24
Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
25
Institut d’Estudis Espacials de Catalunya (IEEC), 08193 Barcelona, Spain
26Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA 27
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
28Centro de Investigaciones Energ´eticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain 29
Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, California 94305, USA
30Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India 31
Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, Illinois 60510, USA
32Excellence Cluster Universe, Boltzmannstrasse 2, 85748 Garching, Germany 33
Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstrasse 1, 81679 Munich, Germany
34Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, California 91109, USA
35Department of Astronomy/Steward Observatory,
933 North Cherry Avenue, Tucson, Arizona 85721-0065, USA
36Department of Astronomy, University of Michigan, Ann Arbor, Michigan 48109, USA 37
Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, United Kingdom
38Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA 39
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
40
Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain
41Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München,
Scheinerstr. 1, 81679 München, Germany
42Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland 43
Santa Cruz Institute for Particle Physics, Santa Cruz, California 95064, USA
44Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA 45
Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany
46Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA 47
Australian Astronomical Observatory, North Ryde, New South Wales 2113, Australia
48Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo,
CP 66318, São Paulo, SP, 05314-970, Brazil
49George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,
and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA
50Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA 51
Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
52Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany 53
Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany
54Department of Physics and Astronomy, Pevensey Building, University of Sussex,
Brighton, BN1 9QH, United Kingdom
55SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 56
ICTP South American Institute for Fundamental Research Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil
57
School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom
58Brandeis University, Physics Department, 415 South Street, Waltham Massachusetts 02453 59
Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil
60
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
(Received 12 October 2018; published 7 June 2019)
We present constraints on extensions of the minimal cosmological models dominated by dark matter and dark energy,ΛCDM and wCDM, by using a combined analysis of galaxy clustering and weak gravitational lensing from the first-year data of the Dark Energy Survey (DES Y1) in combination with external data. We consider four extensions of the minimal dark energy-dominated scenarios: (1) nonzero curvature Ωk,
(2) number of relativistic species Neffdifferent from the standard value of 3.046, (3) time-varying
pivot redshift, wp, and wa), and (4) modified gravity described by the parametersμ0andΣ0that modify the
metric potentials. We also consider external information from Planck cosmic microwave background measurements; baryon acoustic oscillation measurements from SDSS, 6dF, and BOSS; redshift-space distortion measurements from BOSS; and type Ia supernova information from the Pantheon compilation of datasets. Constraints on curvature and the number of relativistic species are dominated by the external data; when these are combined with DES Y1, we findΩk¼ 0.0020þ0.0037−0.0032 at the 68% confidence level, and the
upper limit Neff <3.28ð3.55Þ at 68% (95%) confidence, assuming a hard prior Neff>3.0. For the
time-varying equation-of-state, we find the pivot valueðwp; waÞ ¼ ð−0.91þ0.19−0.23;−0.57þ0.93−1.11Þ at pivot redshift
zp¼ 0.27 from DES alone, and ðwp; waÞ ¼ ð−1.01þ0.04−0.04;−0.28þ0.37−0.48Þ at zp¼ 0.20 from DES Y1 combined
with external data; in either case we find no evidence for the temporal variation of the equation of state. For modified gravity, we find the present-day value of the relevant parameters to beΣ0¼ 0.43þ0.28−0.29 from DES Y1 alone, andðΣ0;μ0Þ ¼ ð0.06þ0.08−0.07;−0.11þ0.42−0.46Þ from DES Y1 combined with external data. These modified-gravity constraints are consistent with predictions from general relativity.
DOI:10.1103/PhysRevD.99.123505
I. INTRODUCTION
Evidence for dark matter[1]and the discovery of cosmic acceleration and thus evidence for dark energy[2,3]were pinnacle achievements of cosmology in the 20th century. Yet because of the still-unknown physical mechanisms behind these two components, understanding them presents a grand challenge for the present-day generation of cos-mologists. Dark matter presumably corresponds to an as-yet undiscovered elementary particle whose existence, along with couplings and other quantum properties, is yet to be confirmed and investigated. Dark energy is even more mysterious, as there are no compelling models aside, arguably, from the simplest one of vacuum energy.
Dark matter and dark energy leave numerous unambigu-ous imprints in the expansion rate of the universe and in the rate of growth of cosmic structures as a function of time. The theoretical modeling and direct measurements of these signatures have led to a renaissance in data-driven cosmol-ogy. Numerous ground- and space-based sky surveys have dramatically improved our census of dark matter and dark energy over the past two decades, and have led to a consensus model with ∼5% energy density in baryons, ∼25% in cold (nonrelativistic) dark matter (CDM), and ∼70% in dark energy. These probes, reviewed in [4–6], include the cosmic microwave background (CMB; [7]); galaxy clustering including the location of the baryon acoustic oscillation (BAO) feature and the impact of redshift space distortions (RSD); distances to type Ia supernovae (SNe Ia); weak gravitational lensing (WL [8]), given by tiny distortions in the shapes of galaxies due to the deflection of light by intervening large-scale structure; and the abundance of clusters of galaxies [9].
The simplest and best-known model for dark energy is the cosmological constant. This model, represented by a single parameter given by the magnitude of the cosmo-logical constant, is currently in good agreement with data. On the one hand, vacuum energy density is predicted to
exist in quantum field theory due to zero-point energy of quantum oscillators, and manifests itself as a cosmological constant: unchanging in time and spatially smooth. On the other hand, the theoretically expected vacuum energy density is tens of orders of magnitude larger than the observed value as has been known even prior to the discovery of the accelerating universe[10,11]. Apart from the cosmological constant, there exists a rich set of other dark energy models including evolving scalar fields, modifications to general relativity, and other physically motivated possibilities[12–14]with many possible avenues to test them with data[15]. Testing for such extensions of the simplest dark energy model on the present-day data has spawned an active research area in cosmology[16–32], and is the subject of the present paper.
The Dark Energy Survey (DES1)[33] is a photometric survey imaging the sky in five filters (grizY) using the 570 Mpixels, 3 deg2 field-of-view Dark Energy Camera (DECam)[34], mounted on the 4-meter Blanco telescope at the Cerro Tololo International Observatory in Chile. After more than five years of data-taking, the survey will end in early 2019 with more than 300 million galaxies catalogued in an area of roughly5000 deg2.
In 2017 the DES collaboration published the analyses of its first year of data (Y1). It presented results which put constraints on certain cosmological parameters derived from their late-universe imprints in galaxy surveys at the same level of precision as the constraints obtained on these same parameters from their early-universe signatures in the CMB data. These results, described in [35] (hereafter Y1KP) are based on the two-point statistics of galaxy clustering and weak gravitational lensing. The combined analysis of the three different two-point correlation func-tions (galaxy clustering, cosmic shear, and the galaxy-shear cross-correlation, typically referred to as galaxy-galaxy
lensing) is the end product of a complex set of procedures which includes the analysis pipeline and methodology[36], its validation on realistic simulations [37], the creation of shape catalogs [38], the estimation and validation of the redshift distribution for different galaxy samples [39], measurement and derivation of cosmological constraints from the cosmic shear signal [40], galaxy–galaxy lensing results [41]and the galaxy clustering statistics [42]. Both alone and in combination with external data from CMB (Planck [43]), BAO (6dF Galaxy Survey [44], the SDSS Data Release 7 Main Galaxy Sample [45], BOSS Data Release 12 [46]) and SNe Ia (Joint Lightcurve Analysis (JLA [47]), DES provides precise measurements in the parameters describing the amplitude of mass fluctuations perturbation and the matter energy density evaluated today. We refer the reader to Y1KP for more details of the DES Y1 analysis, and to Sec.II Dbelow for further description of external data.
In Y1KP we considered only the two simplest models for dark energy: the standard cosmological constant ΛCDM model and a wCDM model with an extra parameter (the dark energy equation-of-state w) accounting for a constant relation between the pressure and the energy density of the dark energy fluid (p¼ wρ). In this paper we explore the impact of the DES Y1 data on the analysis of a few extensions of the standard flatΛCDM and wCDM models considered in Y1KP, namely the possibilities of:
(i) Nonzero spatial curvature;
(ii) New relativistic degrees of freedom;
(iii) Time-variation of the dark energy equation-of-state; (iv) Modifications of the laws of gravity on cosmological
scales.
We describe these extensions in more detail below. Our analysis applies the same validation tests with respect to assumptions about the systematic biases, analysis choices, and pipeline accuracy, as previously done in Y1KP. We also adopt the parameter-level blinding pro-cedure used in that paper, and we do not look at the final cosmological constraints until after unblinding, when the analysis procedure and estimates of uncertainties on various measurement and astrophysical nuisance parameters were frozen. Validation and parameter blinding are also described in further detail below.
Our study effectively complements and extends a num-ber of studies of extensions toΛ=wCDM in the literature using state-of-the-art data, e.g., by Planck [25,43], the Baryon Oscillation Spectroscopic Survey (BOSS)[46], the Kilo Degree Survey (KiDS)[28,48]and more recently by using the Pantheon compilation of SNe Ia data[49]. These studies report no significant deviations from ΛCDM. We will comment on the comparison of our results to these existing constraints in the conclusions.
The paper is organized as follows: the data sets used in the analyses are described in Sec.II, while the models and parameters used to describe the data are detailed in Sec.III.
To ensure that our analysis will not misattribute an astrophysical systematic error to a detection of an exten-sion, we present a series of validation tests in Sec.IV. In Sec.V, we present our results before concluding in Sec.VI.
II. DATA
The primary data used in this study are the auto- and cross-correlations of galaxy positions and shapes measured in data taken by the Dark Energy Survey during its first year of observations.2 We refer the reader to Y1KP for details and only give a summary here.
A. Catalogs
The images taken between August 31, 2013 and February 9, 2014 were processed with the DES Data Management (DESDM) system [50–53], and its outputs validated and filtered to produce the high-quality DES Y1 Gold catalog[54].
From the galaxies in this catalog, we define two samples to be used here: lens galaxies, for which we measure the angular correlation function of positions, and source galaxies, for which we measure the auto-correlation of shapes and the cross-correlation of shapes with lens galaxy positions. To reduce the impact of varying survey character-istics and to remove foreground objects and contaminated regions, we define both samples over an area of1321 deg2. As lens galaxies, we use a sample of luminous red galaxies identified with theREDMAGIC algorithm[55]. This choice is motivated by the small uncertainties in photometric redshifts, high completeness over most of our survey, and the strong clustering of these galaxies. We divide the
REDMAGIC sample into five redshift bins, using three
different cuts on intrinsic luminosity to ensure complete-ness. For bins of redshift z∈½ð0.15−0.3Þ; ð0.3−0.45Þ; ð0.45−0.6Þ, we chose a luminosity cut of L > 0.5Lwith
a spatial density¯n ¼ 10−3 ðh−1MpcÞ−3, where the comov-ing density assumes a fiducialΛCDM cosmology. For the additional redshift bins z∈ ð0.6–0.75Þ and (0.75–0.9), the luminosity cuts and densities are L > L, ¯n ¼ 4 × 10−4 ðh−1MpcÞ−3 and L >1.5L
, ¯n ¼ 10−4 ðh−1MpcÞ−3,
respectively. In total, these samples contain approximately 660,000 lens galaxies.
The primary systematic uncertainties in this catalog are based on residual correlations of galaxy density with observational characteristics of the survey, and in the uncertainty and bias of the lens galaxy redshifts as estimated from the broad-band photometry. The first effect is studied in detail and corrected in [42]. The redshift distributions estimated for the REDMAGIC galaxies are
validated, and the budget for residual uncertainties in
2The DES Y1 data products used in this work are publicly
quantified, using their clustering with spectroscopic galaxy samples [56].
To generate a catalog of source galaxies with accurate shapes for estimating lensing signals, we use the
METACALIBRATION method [57,58] on top of NGMIX.3 NGMIX provides the ellipticity measurements for a suffi-ciently resolved and high signal-to-noise subsample of the Y1 Gold catalog by fitting a simple Gaussian mixture model, convolved with the individual point spread function, to the set of all single exposures taken of a galaxy. The primary systematic uncertainty in this catalog is a multi-plicative error on the mean shear measurement due to biases related to noise and selection effects. In the
METACALIBRATION scheme, this bias is removed by intro-ducing an artificial shear signal and measuring the response of the mean measured ellipticity to the introduced shear. To this end, all galaxy images are artificially sheared, and their ellipticities and all properties used for selecting the sample are remeasured on the sheared versions of their images. By applying a response correction to all estimated shear signals, we find that this method provides measurements with a small multiplicative bias that is dominated by the effect of blending between neighboring galaxies [38].
To divide these source galaxies into redshift bins, we use the means of the redshift probability distributions provided by a version of theBPZ algorithm[59]. This procedure is
based on the METACALIBRATION measurements of griz
galaxy fluxes, as detailed in [39]. By splitting on zmean ∈ ½ð0.2–0.43Þ; ð0.43–0.63Þ; ð0.63–0.9Þ; ð0.9–1.3Þ,
we generate four bins with approximately equal density. The redshift distribution of each source bin is initially estimated from the stack of individual galaxy BPZ redshift probability distributions. This initial estimate is validated, and the systematic uncertainty on the mean redshift in each bin is estimated using a resampling method of high-quality photometric redshifts gained from multiband data in COSMOS [39] and the clustering of the sources with
REDMAGIC galaxies[60,61].
The lens and source galaxy distributions are shown in Fig. 1. The systematic uncertainties on redshift of both samples, and on the shear estimates of the source sample, are quantified in [38,39] and marginalized over in all cosmological likelihoods.
B. Measurements
For the lens and source sample, we use measurements of the three sets of two-point functions in[35]:
(i) Galaxy clustering: the autocorrelation of lens galaxy positions in each redshift bin wðθÞ, i.e., the fractional excess number of galaxy pairs of separation θ relative to the number of pairs of randomly distrib-uted points within our survey mask [42],
(ii) Cosmic shear: the autocorrelation of source galaxy shapes within and between the source redshift bins, of which there are two componentsξðθÞ, taking the products of the ellipticity components of pairs of galaxies, either adding (þ) or subtracting (−) the component tangential to the line connecting the galaxies and the component rotated byπ=4 [40], (iii) Galaxy-galaxy lensing: the mean tangential
elliptic-ity of source galaxy shapes around lens galaxy positions, for each pair of redshift bins,γtðθÞ[41]. Details of these measurements and the checks for potential systematic effects in them are described in detail in[40– 42], and an overview of the full data vector is given in[35]. Here we follow Y1KP, and refer to results from combining all 3 two-point functions as“DES Y1 3 × 2pt.”
Each of these measurements is performed in a set of 20 logarithmic bins of angular separation between 2.5’ and 250’ using the software TREECORR [62]. We only use a
subset of these bins, removing small scales on which our model is not sufficiently accurate. The fiducial scales that we use for clustering and galaxy-galaxy lensing correspond to minimal scale of R¼ 8 h−1Mpc and 12 h−1Mpc, respectively. For cosmic shear, the minimal angular scale θmin is redshift-dependent, and is determined by requiring
that the cross-correlationðξÞijðθminÞ at a pair of redshift
bins i and j not incur an expected fractional contribution from baryonic interactions exceeding 2%; see [40] for details.
For the curvature, number of relativistic species, and dark energy tests, we use the exact same set of scales as in Y1KP, and the data vector with a total of 457 measurements
FIG. 1. Estimated redshift distributions of the lens and source galaxies used in the analysis. The shaded vertical regions define the bins: galaxies are placed in the bin spanning their mean photo-z estimate. We show both the redshift distributions of galaxies in each bin (colored lines) and their overall redshift distributions (black lines).
inðwðθÞ; ξðθÞ; γtðθÞÞ. For our modified gravity tests, we
use a more stringent range of scales, described at the end of Sec. III C 4; this data vector spans only the linear scales, and has a total of 334 measurements.
DES Y1 measurements provide information at z≲ 1, when—in most models—dark energy starts to play a role in cosmic evolution. They provide information about both the geometrical measures (distances, volumes) and the growth of cosmic structure. In particular, both lensing and galaxy clustering are sensitive to the growth of structure, while the kernels in the calculation of the corresponding two-point correlation functions also encode the geometry given by distances (see, e.g., equations in Sec. IV of Y1KP). Therefore, all of the DES Y13 × 2pt measurements probe both geometry and the growth of structure, and thus complement the largely geometrical external data discussed below in Sec.II D. The geometry-plus-growth aspect of the DES Y1 3 × 2pt measurements makes them particularly sensitive to predictions of the models studied in this paper such as modified gravity.
C. Covariance
The statistical uncertainties of these measurements are due to spatial variations in the realizations of the cosmic matter density field (cosmic variance) and random proc-esses governing the positions (shot noise) and intrinsic orientations (shape noise) of galaxies. We describe these uncertainties and their correlations with a covariance matrix C, which is calculated using COSMOLIKE [63] using the relevant four-point functions in the halo model[64]. Shot and shape noise are scaled according to the actual number of source galaxies in our radial bins to account for source clustering and survey geometry. Details of this approach are described in [63,65], along with our validation of the covariance matrix and the corresponding Gaussian likelihood.
D. External data
Combining the DES large-scale structure weak lensing and galaxy clustering data with other, independent probes has benefits in constraining the beyond-minimal cosmo-logical models considered in this paper. In particular, the measurements of distances by SNe Ia and BAO, along with the distance to recombination from the CMB, provide precise geometrical measures, while redshift-space distor-tions (RSD) are sensitive to the growth of cosmic structure [66–70]. These external data significantly complement the combination of geometry and growth probed by the DES clustering and lensing data. Similarly, combining DES with external data enables the comparison of the inferred cosmology from early- and late-time probes (see, e.g., Fig. 11 in Y1KP).
As in Y1KP, we combine DES data with a collection of external data sets to derive the most precise constraints on
the ΛCDM extensions models. We use CMB, CMB
Lensing, BAO, RSD, and Supernova Ia measurements in various combinations. Our final set of external data, described in more detail below, is similar to that used in Y1KP; the main differences are that we add RSD mea-surements from BOSS, and that we update the JLA supernova dataset used in Y1KP to the more recent Pantheon results.
We treat the likelihoods of individual external datasets as independent, simply summing their log-likelihoods. We now describe the individual external datasets that we add to DES data in our combined analysis.
1. CMB & CMB lensing
The cosmic microwave background temperature T and polarization (E- and B-modes) anisotropies are a powerful probe of the early universe. The combination of a rich phenomenology with linear perturbations to a background yields very strong constraints on density perturbations in the early Universe, and on reionization.
In this work we use the Planck 2015 likelihood4 as described in Aghanim et al. [71]. We use the Planck TT likelihood for multipoles30 ≤ l ≤ 2508 and the joint TT, EE, BB and TE likelihood for2 ≤ l ≤ 30. We refer to this likelihood combination as TTþ lowP.5
Planck primary CMB measurements like these strongly constrain all of the baseline cosmological parameters that we use across our models. They have varying power to constrain extension parameters.
We also make use of Planck CMB lensing measurements [72], from temperature only.6 These are measured from higher-order correlations in the temperature field, and act like an additional narrow and very high redshift source sample. We neglect any cross-correlation between DES Y1 measurements and Planck’s CMB lensing map because (1) the noise in the CMB lensing map is sufficiently large; (2) the overlap of the surveys is small compared to the total CMB lensing area; and (3) the CMB lensing autospectrum receives most of its contribution from z≃ 2, while DES constraints are at z≲ 1, which further reduces covariance. We have explicitly tested that the assumption of ignoring the DES-CMBlens covariance holds to an excellent accuracy.
2. BAO + RSD
BAO measurements locate a peak in the correlation function of cosmic structure that corresponds to the sound
4Planck 2018 results[32]were released as this paper was in
advanced stages of the analysis, so we stick with using the Planck 2015 likelihood. The main difference between the two is better measurements of CMB polarization in Planck 2018, resulting in better constraints on the optical depthτ.
5We used the public Planck likelihood files
PLIK_LITE_V18_TT. CLIKandLOWL_SMW_70_DX11D_2014_10_03_V5C_AP.CLIK.
6We use the file
horizon at the drag epoch. Since the sound speed before that point depends only on the well-understood ratio of photon to baryon density, this horizon acts as a standard ruler and can be used to measure the angular diameter distance with a percent-level precision.
As in Y1KP we use BAO measurements from BOSS Data Release 12 [46], which provides measurements of both the Hubble parameter HðziÞ and the comoving angular
diameter distance dAðziÞ, at three separate redshifts, zi¼ f0.38; 0.51; 0.61g. The other two BAO data that we use, 6DF Galaxy survey [44] and SDSS Data Release 7 Main Galaxy Sample [45], are lower signal-to-noise and can only tightly constrain the spherically averaged combination of transverse and radial BAO modes, DVðzÞ ≡ ½czð1 þ zÞ2D2AðzÞ=HðzÞ1=3. These constraints
are at respective redshifts z¼ 0.106 (6dF) and z ¼ 0.15 (SDSS MGS).
We also utilize the redshift-space distortion measure-ments from BOSS DR12; they are given as measuremeasure-ments of the quantity fðziÞσ8ðziÞ at the aforementioned three
redshifts. Here f is the linear growth rate of matter perturbations andσ8is the amplitude of mass fluctuations on scales8h−1 Mpc. We employ the full covariance, given by[46], between these three RSD measurements and those of BAO quantities HðziÞ and dAðziÞ. We treat the 6dF
and SDSS MGS measurements as independent of those from BOSS DR12, and we neglect any cosmological dependence on the derived values of fðziÞσ8ðziÞ from
BOSS DR12 data.
Finally, we ignore the covariance between these BAO/ RSD measurements and those of DES galaxy clustering and weak lensing; the two sets of measurements are carried out on different areas on the sky and the covariance is expected to be negligible.
3. Supernovae
Type Ia supernovae (SNe Ia) provide luminosity dis-tances out to redshift of order unity and beyond, and thus excellent constraints on the expansion history of the universe. In this analysis we use the Pantheon SNe Ia sample [49] which combines 279 SNe Ia from the Pan-STARRS1 Medium Deep Survey (0.03 < z < 0.68) with SNe Ia from SDSS, SNLS, various low-z and HST samples. The Pantheon data was produced using the Pan-STARRS1 Supercal algorithm [73] which established global calibration for 13 different SNe Ia samples. The final Pantheon sample includes 1048 objects in the redshift range0.01 < z < 2.26.
III. THEORY AND MODELING A. Standard cosmological parameters
We assume the same set of ΛCDM cosmological parameters described in Y1KP, then supplement it with parameters alternately describing four extensions. We
parametrize the matter energy density today relative to the critical densityΩm, as well as that of the baryonsΩband
of neutrinosΩν.7Moreover, we adopt the amplitude Asand
the scalar index ns of the primordial density perturbations
power spectrum, as well as the optical depth to reionization τ, and the value of the Hubble parameter today H0. Except
in the case of varying curvature, we assume that the universe is flat and, except in the case of varying dark energy, we assume that it is Λ-dominated with w ¼ −1; under those two assumptions, ΩΛ¼ 1 − Ωm. Note that
the amplitude of mass fluctuationsσ8is a derived param-eter, as is the parameter that decorrelates σ8 and Ωm, S8≡ σ8ðΩm=0.3Þ0.5. The fiducial parameter set is therefore θbase ¼ fΩm; H0;Ωb; ns; As;ðτÞg; ð1Þ
where the parentheses around the optical depth parameter indicate that it is used only in the analysis combinations that use CMB data.
To model the fully nonlinear power spectrum, we first estimate the linear primordial power spectrum on a grid of ðk; zÞ usingCAMB[74] orCLASS[75]. We then apply the HALOFIT prescription [76–78] to get the nonlinear
spec-trum. Throughout this work, we employ the version from Takahashi et al.[77].
In addition to this set ofΛCDM parameters, we use the following parametrization for each of the extension models:
(1) Spatial curvature:Ωk;
(2) The effective number of neutrinos species Neff; (3) Time-varying equation-of-state of dark energy:
w0, wa;
(4) Tests of gravity:ΣðaÞ, μðaÞ.
We describe these extensions in more detail below in Sec.III C.
B. Nuisance parameters
We follow the analysis in Y1KP, and model a variety of systematic uncertainties using an additional 20 nuisance parameters. The nuisance parameters are
(i) Five parameters bi that model linear bias of lens galaxies in five redshift bins;
(ii) Two parameters, AIAandηIA, that model the power spectrum of intrinsic alignments as a power-law scaling AIAð1þz1þz0ÞηIA, with z0¼ 0.62 (see Sec. VII B
of[40] for a complete description of the model); (iii) Five parametersΔzi
lto model the uncertainty in the
means of distributions nðziÞ of galaxies in each of
the lens bins; (iv) Four parametersΔzi
sto model the uncertainty in the
means of distributions nðziÞ of galaxies in each of
the source bins;
7In theΣ
0,μ0systematic tests that use the olderMGCAMB, this
was not implemented, soΩνis fixed in these tests. We do varyΩν in our runs on the real data.
(v) Four parameters mi that model the overall uncer-tainty in the multiplicative shear bias in each of the source bins.
All of the cosmological and nuisance parameters in our standard analysis, along with their respective priors, are given in TableI.
Note that we did not change any assumptions about the nuisance parameters relative to our previous analysis applied to ΛCDM and wCDM. It is possible in principle that extensions (e.g., modified gravity) to these simplest models warrant more complicated modeling and therefore more nuisance parameters (e.g., adopting more complicated parametrizations of galaxy bias). To address this possibility, we consider a number of more complicated parametriza-tions of the systematic effects (described in Sec.IV) with the aim of determining whether we could misidentify a systematic effect as evidence for an extension. Our tests, also described in that section, indicate that constraints on the key extension parameters studied in this paper are not
sensitive to these additional parameters. This justifies our choice not to modify our fiducial nuisance parametrization described in the bullet-point list above and used previously in Y1KP. Future, more precise data will require revisiting these, in addition to potentially extracting information about these extensions from the modified behavior of astrophysical nuisance effects.
C. ΛCDM extensions
We now introduce the four extensions to the simplest Λ=wCDM models that we study in this paper. The cosmo-logical parameters describing these extensions, along with priors given to them in our analysis, are given in TableII.
1. Spatial curvature
Standard slow-roll inflation predicts that spatial curva-ture is rapidly driven to zero. In this scenario, the amount of curvature expected today is Ωk≃ 10−4, where the tiny
deviation from zero is expected from horizon-scale per-turbations but will be very challenging to measure even with future cosmological data[79]. Departures from near-zero curvature are however expected in false-vacuum inflation, as well as scenarios that give rise to bubble collisions[80,81]. With curvature, and ignoring the radi-ation density whose contribution is negligible in the late universe, the Hubble parameter generalizes to
HðaÞ
H0 ¼ ½Ωma
−3þ ð1 − Ω
m− ΩkÞ þ Ωka−21=2: ð2Þ
so thatΩk<0 corresponds to spatially positive curvature,
and the opposite sign to the spatially negative case. In this work, we compare constraints onΩkusing DES data alone,
as well as with combinations of subsets of the external data described in Sec.II D.
We do not modify the standard HALOFIT prescription
[76,77]for prediction of the nonlinear power spectrum for nonzero values of Ωk. Simulation measurements of the
nonlinear spectrum for nonzero values ofΩkdo not exist to
TABLE II. Summary of the extensions to the ΛCDM model that we study in this paper, the parameters that describe these extensions, and the (flat) priors given to these parameters. In addition to the priors listed in the table, we also impose the prior w0þ wa≤ 0 for dark energy, and 2Σ0þ 1 > μ0 for modified
gravity.
ΛCDM Extension Parameter Flat Prior
Curvature Ωk ½−0.25; 0.25
Number relativistic species Neff [3.0, 7.0]
Dynamical dark energy w0 ½−2.0; −0.33 wa ½−3.0; 3.0
Modified gravity Σμ0 ½−3.0; 3.0
0 ½−3.0; 3.0
TABLE I. Parameters and priors used to describe the measured two-point functions, as adopted from Y1KP. Flat denotes a flat prior in the range given while Gauss(μ, σ) is a Gaussian prior with meanμ and width σ.
Parameter Prior Cosmology Ωm Flat (0.1, 0.9) As Flat (5 × 10−10, 5 × 10−9) ns Flat (0.87, 1.07) Ωb Flat (0.03, 0.07) h Flat (0.55, 0.91) Ωνh2 Flat (5 × 10−4,10−2)
Lens Galaxy Bias
bi (i¼ 1, 5) Flat (0.8, 3.0)
Intrinsic Alignment AIAðzÞ ¼ AIA½ð1 þ zÞ=1.62ηIA
AIA Flat (−5, 5)
ηIA Flat (−5, 5)
Lens photo-z shift (red sequence) Δz1 l Gauss (0.008, 0.007) Δz2 l Gauss (−0.005, 0.007) Δz3 l Gauss (0.006, 0.006) Δz4 l Gauss (0.000, 0.010) Δz5 l Gauss (0.000, 0.010)
Source photo-z shift Δz1 s Gauss (−0.001, 0.016) Δz2 s Gauss (−0.019, 0.013) Δz3 s Gauss (þ0.009, 0.011) Δz4 s Gauss (−0.018, 0.022) Shear calibration mi METACALIBRATION (i¼ 1, 4) Gauss (0.012, 0.023)
sufficiently validate this regime. However, it is not an unreasonable a priori assumption that the nonlinear modi-fication to the power spectrum is only weakly affected by curvature beyond the primary effect captured in the linear power spectrum being modified. We do incorporate the impact ofΩkin the evolution of the expansion and growth, which is properly modeled as part of the linear matter power spectrum that is modified byHALOFIT. We verify that
this approximation does not significantly impact our results by comparing to the case where we restrict our data to scales that are safely“linear” as described in Sec.IVbelow.
2. Extra relativistic particle species
Anisotropies in the CMB are sensitive to the number of relativistic particle species. The Standard Model of particle physics predicts that the three left-handed neutrinos were thermally produced in the early universe and their abun-dance can be determined from the measured abunabun-dance of photons in the cosmic microwave background. If the neutrinos decoupled completely from the electromagnetic plasma before electron-positron annihilation, then the abundance of the three neutrino species today would be
n¼ Neff×113 cm−3 ð3Þ
with Neff ¼ 3. In actuality, the neutrinos were slightly
coupled during e annihilation, so Neff ¼ 3.046 in the standard model [82–84]. Values of Neff larger than this would point to extra relativistic species. The DES obser-vations are less sensitive to Neffthan the CMB, because the effect of this parameter in the DES mainly appears via the change in the epoch of matter-radiation equality. Nevertheless, DES might constrain some parameters that are degenerate with Neff so, at least in principle, adding
DES observations to other data sets might provide tighter constraints.
There are well-motivated reasons for exploring possibil-ities beyond the standard scenario. First, the most elegant way to obtain small neutrino masses is the seesaw model [85], which typically relies on three new heavy Standard Model singlets, or sterile neutrinos. While these often are unstable and have very large masses, it is conceivable that sterile neutrinos are light and stable on cosmological timescales[86]. Indeed, there are a variety of experimental anomalies that could be resolved with the introduction of light sterile neutrinos, and a keV sterile neutrino remains an interesting dark matter candidate. If one or more light sterile neutrinos do exist, then they would typically be produced in the early universe via oscillations from the thermalized active neutrinos with an abundance determined by the mixing angles. As an example, the LSND/ Miniboone anomaly[87,88]could be resolved with a light sterile neutrino thus implying Neff≃ 4; the mixing angle of the sterile neutrino would dictate that it would have the same abundance as the 3 active neutrinos. More generally, a
wide variety of extensions to the Standard Model contain light stable particles that would have been produced in the early Universe[89] and impacted the value of Neff. It is
important to note that while the addition of an extra relativistic species would explain some aspects of these observations, it is difficult for such models to accommodate all of the existing neutrino oscillation observations.
In the fiducial model, we are allowing for a single free parameterPmν, treating the 3 active neutrinos as degen-erate (since they would be approximately degendegen-erate if they had masses in the range we can probe, >0.1 eV). There is some freedom in how to parametrize the extension of a light sterile neutrino, however. If we attempt to model the addition of a single sterile neutrino, then in principle two new parameters must be added. For example, if the sterile neutrino has the same temperature as the active neutrinos, then the parameters can be chosen to be Neff,
allowed to vary between 3.046 and 4.046, and ms, the mass
of the sterile neutrino. Two light sterile neutrinos would require two more parameters, etc. However, we expect that the cosmological signal will be sensitive primarily to the total neutrino mass density and the number of effective massless species at the time of decoupling, as captured by Neff, so we use only these two parameters,
P
mνand Neff.
Note that a value of Neffappreciably different than 3 would
point to a sterile neutrino or another light degree of freedom. We give Neff a flat prior in the range [3.0,
9.0], where the lower hard bound encodes the guaranteed presence of at least three relativistic neutrino species.
When varying Neff, the fraction of baryonic mass in
helium Yp is set by a fitting formula based on the
PARTHENOPE BBN code [90]. This interpolates a Yp
for a given combination ofΩbh2 and Neff. An additional
prior of Ωbh2<0.04 is applied in the Neff analysis to
restrict the interpolation to its valid range.
3. Time-varying equation-of-state of dark energy Given the lack of understanding of the physical mecha-nism behind the accelerating universe, it is important to investigate whether the data prefer models beyond the simplest one, the cosmological constant. In Y1KP, we investigated the evidence for a constant equation-of-state parameter w≠ −1. We found no evidence for w ≠ −1, with a very tight constraint from the combination of DES Y1, CMB, SNe Ia, and BAO of w¼ −1.00þ0.05−0.04.
We now investigate whether there is evidence for the time-evolution of the equation-of-state w. We consider the phenomenological model that describes dynamical dark energy[91]
wðaÞ ¼ w0þ ð1 − aÞwa; ð4Þ
where w0 is the equation-of-state today, while wa is its variation with scale factor a. Theðw0; waÞ parametrization fits many scalar field and some modified gravity expansion
histories up to a sufficiently high redshift, and has been used extensively in past constraints on dynamical dark energy.
It is also useful to quote the value of the equation-of-state at the pivot wp≡ wðapÞ; this is the scale factor at which the equation-of-state value and its variation with the scale factor are decorrelated, and where wðaÞ is best-determined. Rewriting Eq. (4) as wðaÞ ¼ wpþ ðap− aÞwa, the pivot
scale factor is
ap¼ 1 þ
Cw0wa
Cwawa
ð5Þ where C is the parameter covariance matrix in the 2D ðw0; waÞ space, obtained by marginalizing the full 28 × 28
covariance over the remaining 26 parameters. The corre-sponding pivot redshift is of course zp¼ 1=ap− 1.
The linear-theory observable quantities in this model are straightforwardly computed, as the new parameters affect the background evolution in a known way, given that the Hubble parameter becomes
HðaÞ
H0 ¼ ½Ωma
−3þ ð1 − Ω
mÞa−3ð1þw0þwaÞe−3wað1−aÞ1=2:
ð6Þ To obtain the nonlinear clustering in theðw0; waÞ model,
we assume the same linear-to-nonlinear mapping as in the ΛCDM model, except for the modified expansion rate HðzÞ [92–94]. In particular, we implement the same HALOFIT
nonlinear [76,77] prescription as we do in the fiducial ΛCDM case. We impose a hard prior w0þ wa≤ 0; models
lying in the forbidden region have a positive equation of state in the early universe, are typically ruled out by data, and would present additional challenges in numerical calculations. For the same reason we impose the prior w0<−0.33. Note also that in our analysis we do implicitly allow the“phantom” models where wðaÞ < −1; while not a feature of the simplest physical models of dark energy (e.g., single-field quintessence), such a violation of the weak energy condition is in general allowed[95].
4. Modified gravity
The possibility of deviations from general relativity on cosmological scales has been motivated by the prospect that an alternative theory of gravity could offer an explanation for the accelerated expansion of the Universe. In the past several years, numerous works constraining modifications to gravity using cosmological data have been published, including from the Planck team [25,32], the Kilo Degree Survey [28], and the Canada-France-Hawaii Lensing Survey [96]. Constraints from the Dark Energy Survey Science Verification data were obtained in[97]. Recently, stringent constraints were made on certain alternative theories of
gravity[98–104]via the simultaneous observation of gravi-tational and electromagnetic radiation from a binary neutron star merger with the Laser Interferometer Gravitational Wave Observatory (LIGO)[105].
In what follows, we refer to the scalar-perturbed Friedmann-Robertson-Walker line element in the con-formal Newtonian gauge:
ds2¼ a2ðτÞ½ð1 þ 2ΨÞdτ2− ð1 − 2ΦÞδijdxidxj: ð7Þ In general relativity and without anisotropic stresses, Ψ ¼ Φ. The parametrization of deviations from General Relativity studied in this work is motivated by theoretical descriptions which make use of the quasistatic approxi-mation (see, e.g.,[106]). It can be shown that in the regime where linear theory holds and where it is a good approxi-mation to neglect time derivatives of novel degrees of freedom (e.g., extra scalar fields), the behavior of the majority of cosmologically motivated theories of gravity can be summarized via a free function of time and scale multiplying the Poisson equation, and another which represents the ratio between the potentials Φ and Ψ. Such a parametrization is an effective description of a more complicated set of field equations[107–116], but this approximation has been numerically verified on scales relevant to our present work[117–121].
There are a number of related pairs of functions of time and scale which can be used in a quasistatic parametrization of gravity; we choose the functionsμ and Σ, defined as
k2Ψ ¼ −4πGa2ð1 þ μðaÞÞρδ; ð8Þ
k2ðΨ þ ΦÞ ¼ −8πGa2ð1 þ ΣðaÞÞρδ; ð9Þ where we are working in Fourier space where k is the wave number, andδ is the comoving-gauge density perturbation. This version of the parametrization was used in[25,32,96], and benefits from the fact thatΣ parametrizes the change in the lensing response of massless particles to a given matter field, whileμ is linked to the change in the matter overdensity itself. Therefore, weak lensing measurements are primarily sensitive to Σ but also have some smaller degree of sensitivity toμ via their tracing of the matter field, whereas galaxy clustering measurements depend only onμ and are insensitive to Σ. We find the DES data alone are more sensitive to Σ than to μ; constraining the latter requires combining DES with a nonrelativistic tracer of large-scale structure such as the RSD (e.g.,[96,97]) which we also do below as part of our combined analysis.
To practically constrainμ and Σ, we select a functional form of
μðzÞ ¼ μ0ΩΩΛðzÞ Λ
; ΣðzÞ ¼ Σ0ΩΛðzÞ
whereΩΛðzÞ is the redshift-dependent dark energy density (in theΛCDM model) relative to critical density, and ΩΛis its value today. This time dependence has been introduced in[122], and is widely employed (see, e.g.,[25,32,96]). It is motivated by the fact that in order for modifications to GR to offer an explanation for the accelerated expansion of the Universe, we would expect such modifications to become significant at the same timescale as the acceleration begins. We do not model any scale-dependence ofμ and Σ since it has been shown to be poorly constrained by current cosmological data while not much improving the good-ness-of-fit [25]. We therefore include only the parameters μ0 and Σ0 (but, as explained in Sec. IVA, only quote
constraints onΣ0). In GR,μ0¼ Σ0¼ 0.
Note that although our choice of parametrization is motivated by the quasistatic limit of particular theories of gravity, our analysis takes an approach which is completely divorced from any given theory. We endeavor instead to make empirical constraints on the parametersμ0andΣ0as specified by Eqs.(8), (9), and(10). Because we take this empirically driven approach, we include certain data ele-ments in which the quasistatic approximation would not be expected to hold, most importantly the near-horizon scales for the Integrated Sachs-Wolfe (ISW) effect. Although not rigorously theoretically justified, a similar approach with respect to inclusion of the ISW effect at large scales was taken in, e.g.,[96]. Practically, this choice has the benefit of providing an important constraint onτ from external CMB data, which is useful in breaking degeneracies.
We use COSMOSIS with a version of MGCAMB8
[123,124] modified to include the Σ, μ parametrization to compute the linear matter power spectrum and the CMB angular power spectra. For some sets of (Σ0,μ0) MGCAMB
returns an error; we estimated this region of parameter space can be avoided by imposing an additional hard prior μ0<1 þ 2Σ0. We therefore implement this prior in order to
avoid computations for parameters not handled by MGCAMB. The effects of this hard prior can be visually
observed in our constraints on modified-gravity parameters with DES alone (Fig. 4 below); we demonstrate in the Appendix that its effects on the combined DESþ external constraint is likely to be minimal.
To validate our modified-gravity analysis pipeline, we compare the COSMOSIS results to that of another code,
COSMOLIKE[63]. We require that the two codes give the
same theory predictions for clustering and lensing observ-ables, and the same constraints on cosmological parameters given a synthetic data vector. The comparison shows good agreement, and details can be found in Appendix.
Finally, because the ðμ; ΣÞ description does not con-stitute a complete theoretical model, its nonlinear clustering predictions are not available to us even in principle. We therefore restrict ourselves to the linear-only analysis. To do
this, we follow the Planck 2015 analysis[25]and consider the difference between the nonlinear and linear-theory predictions in the standardΛCDM model at best-fit values of cosmological parameters and with no modified gravity. Using the respective data vector theory predictions, dNL
and dlin, and full error covariance of DES Y1, C, we
calculate the quantity Δχ2≡ ðd
NL− dlinÞTC−1ðdNL− dlinÞ ð11Þ
and identify the single data point that contributes most to this quantity. We remove that data point, and repeat the process untilΔχ2<1. The resulting set of 334 (compared to the original 457) data points that remain constitutes our fiducial choice of linear-only scales.
IV. VALIDATION TESTS AND BLINDING We subject ourΛCDM extensions analyses to the same battery of tests for the impact of systematics as in Y1KP. The principal goal is to ensure that all of our analyses are robust with respect to the effect of reasonable extensions to models of astrophysical systematics and approximations in our modeling. As part of the same battery of tests, we also test that the range of spatial scales that are used lead to unbiased cosmological results, and that motivated mod-ifications to our modeling assumptions do not significantly change the inferred cosmology.
In these tests and the results below,9 sampling of the posterior distribution of the parameter space is performed with MULTINEST[125]andEMCEE [126]wrappers within COSMOSIS10[127]and COSMOLIKE[63]. While the
con-vergence of MULTINEST is intrinsic to the sampler and
achieved by verifying that the uncertainty in the Bayesian evidence is below than some desired tolerance, we explic-itly check the convergence ofEMCEEchains. In order to do so, we compute the autocorrelation length of each walk, then continue the walks until a large number of such lengths is reached.11The autocorrelation length estimates how long a chain needs to be in order for new “steps” to be uncorrelated with previous ones. We then split chains into several uncorrelated segments and verify that marginalized parameter constraints do not change significantly when
8https://aliojjati.github.io/MGCAMB/mgcamb.html
9One important distinction from the data-based results in later
sections is that we sample a lower-precision version of the CMB lensing contribution to constraints including external data when varying Ωk, then modify the posterior to the higher precision
prediction via importance sampling. We do this to speed up the evaluation of nonflat models, as in our implementation of CAMB, particularly when evaluating CMB lensing, sampling over many chains at full precision is impractical. We checked that this approximation has a minor effect on the shape of the posterior.
10https://bitbucket.org/joezuntz/cosmosis/
11The recommended methods for convergence testing (as well
as the documentation forEMCEE) can be found inhttps://emcee .readthedocs.io/
these segments are compared with each other. The typical number of samples of the posterior in these chains is between two and three million. We have also verified in select cases that this procedure leads to excellent agreement with the 1D marginalized parameter posteriors achieved by MULTINEST, so both samplers are used interchangeably in
what follows.
A. Validation of assumptions using synthetic data In order to verify that our results are robust to modeling assumptions and approximations, we compare the inferred values of the extension parameters (Ωk; Neff;…) obtained by a systematically shifted, noiseless synthetic data vector. The synthetic data vector is centered precisely on the standard ΛCDM cosmology, except it is shifted with the addition of a systematic effect that is not included in our analysis. The goal is to ensure that we do not claim evidence for an extension to ΛCDM when the real data contains astrophysical effects more complex than those in our model. For each systematic effect, we compare the inferred set of extension parameters to the fiducial, unmodified extension parameters used to create the synthetic data (which we refer to as the“baseline” constraint). For all of these tests, for DES we use the synthetic data vectors (for the baseline case and the systematic shifts described below), but for the external data sets—CMB, BAO, RSD, and SN Ia—we use the actual, observed data vector.
The changes to modeling assumptions that we con-sider are
(1) Baryonic effects: we synthesize a noiseless data vector including a contribution to the nonlinear power spectrum caused by AGN feedback using the OWLS AGN hydrodynamical simulation [128] and following the methodology of [40].
(2) Intrinsic alignments, simple case: we synthesize a noiseless data vector with the IA amplitude AIA ¼ 0.5 and redshift scaling ηIA ¼ 0.5 using the baseline nonlinear alignment model used in Y1KP. While we explicitly marginalize over these IA parameters in our analysis, this systematic check is still useful to monitor any potential biases due to degeneracy between the cosmological parameters and ðAIA;ηIAÞ and the presence of non-Gaussian posteriors.
(3) Intrinsic alignments, complex case: we synthesize a noiseless data vector using a subset of the tidal alignment and tidal torquing model (hereafter TATT) from[129]. This introduces a tidal torquing term to the IA spectrum that is quadratic in the tidal field. The TATT amplitudes were set to A1¼ 0, A2¼ 2 with no z dependence, as was done in[130] when validating the analysis of Y1KP.
(4) Nonlinear bias: we test our fiducial linear-bias assumption by synthesizing a noiseless data vector that models the density contrast of galaxies as
δg ¼ bi1δ þ
1
2bi2½δ2− σ2 ð12Þ
whereδ and δg are the overdensities in matter and galaxy counts respectively, and the density variance σ2¼ hδ2i is subtracted to enforce hδ
gi ¼ 0. While
this relationship is formally defined for smoothed density fields, the results do not depend on the choice of smoothing scale since, e.g., the variance explicitly cancels with contributions to the two-point correlation. We are considering scales that are suffi-ciently larger than the typical region of halo formation that we neglect higher-derivative bias terms. See Ref.[131]for further discussion of nonlinear biasing. Here i refers to the lens redshift bin and where bi1¼ f1.45; 1.55; 1.65; 1.8; 2.0g for the five bins. The b2
values used for each lens bin were estimated from the following relationship fit in simulations [132]: b2¼ 0.412–2.143b1þ 0.929b21þ 0.008b31. Because the contribution from tidal bias bs2is expected to be
small, we set it to zero in these validation tests. (5) Magnification: we synthesize a noiseless data vector
that includes the contribution from magnification to γt and wðθÞ. These are added in Fourier space
using[133].
(6) Limber approximation and RSD: we synthesize a noiseless data vector that uses the exact (non-Limber) wðθÞ calculation12
and include the contri-bution from redshift space distortions[135]. More information about the implementation of these tests can be found in[130].
The results of these tests are shown in Fig. 2. The columns show the parameters describing ΛCDM exten-sions, namely wp, wa, Ωk, Neff, Σ0, and μ0. The shaded
vertical region shows the marginalized 68% posterior confidence limit (CL) in each parameter for the baseline case. The horizontal error bars show how this posterior, fully marginalized over all other parameters, including the other parameter in two-parameter extensions, changes with the systematic described in the given row for the case of DES-only (blue bars) and DESþ external (red bars) data. We observe that, except in the cases explained below, the marginalized posteriors are consistent with the baseline analysis in these tests.13
Figure 2 shows shifts in some DES-only 68% C.L. constraints relative to the input value shown by the dotted vertical lines. The most pronounced effect is in the
12We do not investigate the effect of the Limber approximation
on the tangential shear profileγt, since it includes the projection
from the observer to the source galaxy, and is less sensitive to the Limber approximation, below the level of the DES Y1 statistical uncertainty[134].
13The DESþ external Ω
kcolumn in Fig.2is narrow and hard
to visually inspect, but we have verified that there are no biases in the curvature parameter with alternate assumptions about the systematic errors shown in the different rows.
DES-only case for modified gravity parameterμ0(and, to a slightly smaller extent,Σ0and Neff), which is more than1-σ away from its true value of zero. Upon investigating this, we found that the bias away from the input value is caused by the interplay of two effects: (1) weak constraints, with a relatively flat likelihood profile in these parameters in certain directions, combined with (2) prior-volume effect, where the large full-parameter-space volume allowed in the direction in which the parameter is a reasonably good fit ends up dominating the total integrated posterior, resulting in a 1D marginalized posterior that is skewed away from the maximum likelihood true value. For example, with the restricted range of scales that we use for the modified gravity tests, negative values of μ0 are an acceptable (though not the best) fit and, because of the relatively large number of combinations of other parameters that result in a good likelihood for−3 ≤ μ0≲ 0, the 68% C.L. constraint on μ0 ends up excluding the input best-fit value of zero (see Fig. 2). We have explicitly checked that removing the principal degeneracy with other parameters—in modified gravity tests, achieved by fixing the bias parameters bi—removes the bias in μ0. Nevertheless,
because these tests imply that the DES-only constraint on this
parameter would suffer from the aforementioned bias, we choose not to quote constraints onμ0from the DES-only data in the results below.
We also observe a bias in the DESþ external constraint on wa relative to the input value of zero. This is mostly
driven by the fact that the best fit of the external data does not necessarily coincide with the cosmological parameter values assumed for the synthetic data vectors used to produce DES constraints—in fact, it is well-known that external data alone favor wa <0[32]. Additionally, even
the DES synthesized data alone mildly prefer negative wa due to the prior-volume effect mentioned above. The resulting synthesized DESþ external constraint on wa is
then biased negative at greater than 68% confidence. Because the combined analysis on the real data will not be subject to the principal cause of the wa bias observed here, we proceed with the analysis.
There are therefore two takeaways from Fig. 2: (i) First, the projected 1D inferences from DES-only
measurements onμ0are likely to be biased princi-pally due to the prior volume effect, so we choose not to quote constraints on this parameter in the DES-only case (but still include it in the analysis
FIG. 2. Impact of assumptions and approximations adopted in our analysis, demonstrated on synthetic data (that is, noiseless DES data centered on the theoretical expectation, along with actual external data). Each column shows one of the cosmological parameters describingΛCDM extensions; the dotted vertical line is the true input value of that parameter in the DES data vector (which does not necessarily coincide with the parameter values preferred by the external data). The vertical shaded bands show the marginalized 68% CL constraints in the baseline model for the DES-only synthetic data (blue) and DESþ external. The horizontal error bars show the inferred constraint for each individual addition to the synthetic data vector which are listed in rows; they match the shaded bands for the baseline case. For subsequent rows, they show the inferred constraint for each individual addition to the synthetic data vector as listed on the right. Some cases that appear inconsistent with the baseline analysis are discussed further in Sec.IVA. In cases where the prior is informative, we also include a dashed vertical line to signify the prior edge.
throughout). We do not attempt to correct the biases in the w0-wa DESþ external case or inflate the
parameter errors to account for it; see the discus-sion above.
(ii) Second and most importantly, the different assump-tions considered in Fig.2produce consistent results with the baseline constraint for all parameters describing ΛCDM extensions.
B. Validation of assumptions using DES data In addition to the tests in the previous section that constrain potential biases due to our modeling assumptions and approximations on synthesized noiseless data, we implement several validation tests that modify how we analyze the actual DES data vector. In particular, we test the following assumptions:
(7) Intrinsic alignments, free redshift evolution: while the fiducial analysis assumes IA to scale as a power-law in redshift (see Sec.III B), we relax that here by assuming four uncorrelated constant amplitudes per source redshift bin.
(8) Conservative scales: to gauge how our results depend on the range of angular scales used, we adopt the conservative set of (basically linear) scales used in the modified gravity extension, and apply it to the other three extensions (curvature, Neff, dynamical dark energy).
(9) Alternate photometric redshifts: to investigate the robustness of our results to the shape of the redshift distribution of source galaxies, we adopt the dis-tributions obtained directly from resampling the COSMOS data, as described in[39].
For each of these alternate analysis options, we inves-tigate how the fiducial constraints on theΛCDM extensions parameters change. These results are presented and dis-cussed along with our main results, near the end of Sec.V.
C. Blinding
We follow the same strategy as in Y1KP, and blind the principal cosmological results to protect against human bias. We do so by shifting axes in all plots showing the cosmological parameter constraints. Where relevant, this includes simultaneously not plotting theory predictions (including simulation outputs as “theory”) in those same plots. A different shift was applied to each of the DES, external data, and joint constraint contours in any figures made at the blinded stage. Moreover evidence ratios of the joint constraints were not read before unblinding. This was done to prevent confirmation bias based on the level of agreement between the DES and external constraints.
We unblinded once we ensured that there are no biases on the extension parameters due to systematics, as shown in Figs.2and7, apart from those that have a known, statistical explanation (see Sec.IVA).
We have made two modifications to the analysis after the results were unblinded. First, we identified that the incor-rect Planck data file (PLIK_LITE_V18_TTTEEE.CLIK) was
used for ourðw0; waÞ results and reran these chains with the
correct file (PLIK_LITE_V18_TT.CLIK). We verified that this
modification does not lead to appreciable differences in the final constraints, though it does lead to a difference in the reported Bayesian evidence ratios for this case. Second, we adopted the GETDIST code to evaluate the marginalized
posteriors, as it is more suitable to handle boundary effects in the posteriors[136]. This leads to small differences in cases where the constraints are strongly informed by the prior boundaries, such as Neff.
V. RESULTS
The constraints on curvature and the number of relativ-istic species are given in the two panels of Fig. 3. For curvature, we find
Ωk¼ 0.16þ0.09−0.14 DES Y1
¼ 0.0020þ0.0037
−0.0032 DES Y1 þ External ð13Þ
while for the number of relativistic species, the lower limit hits against our hard prior of Neff >3.0 so we quote only
the 68% (95%) upper limits Neff <5.28ð—Þ DES Y1
<3.28ð3.55Þ DES Y1 þ External; ð14Þ where the dashes indicate that we do not get a meaningful upper limit from DES alone at the 95% since the constraint hits against the upper limit of our prior.
Figure 3indicates that DES alone constrains curvature weakly, showing mild (∼1-σ) preference for positive values ofΩk; note also that this constraint is informed by the upper prior boundary. The DES-only constraint on Neff is also relatively weak, and is fully consistent with the theoreti-cally favored value Neff ¼ 3.046. Moreover, the DES Y1 data do not appreciably change the existing external-data constraints on these two parameters. The addition of the DES data to external measurement does slightly suppress Neff, which can be understood as follows. The DES data
prefer a lowerΩmthan the external data, leading to a slight
increase in h such that the posterior distribution inΩmh3is
downweighted at the high values of this parameter combi-nation. BecauseΩmh3is highly correlated with Neff—they
both generate out-of-phase changes in the CMB temper-ature power spectrum—adding DES to external data also has the consequence of slightly suppressing Neff.
We also compare the cases where the number of relativistic species is fixed at Neff ¼ 3.046 (the standard
model) and Neff ¼ 4.046 (standard model, plus a single fully thermalized sterile neutrino). Preference for one model over the other is assessed using the evidence ratio,
